surreal number

The surreal numbers are a generalization of the reals. Each surreal number consists of two parts (called the left and right), each of which is a set of surreal numbers. For any surreal number N, these parts can be called NL and NR. (This could be viewed as an ordered pair of sets, however the surreal numbers were intended to be a basis for mathematics, not something to be embedded in set theory.) A surreal number is written N=⟨NL∣NR⟩.

Not every number of this form is a surreal number. The surreal numbers satisfy two additional properties. First, if x∈NR and y∈NL then x≰y. Secondly, they must be well founded. These properties are both satisfied by the following construction of the surreal numbers and the ≤relation by mutual induction:

⟨∣⟩, which has both left and right parts empty, is 0.

Given two (possibly empty) sets of surreal numbers R and L such that for any x∈R and y∈L, x≰y, ⟨L∣R⟩.

Define N≤M if there is no x∈NL such that M≤x and no y∈MR such that y≤N.

This process can be continued transfinitely, to define infinite and infinitesimal numbers. For instance if ℤ is the set of integers then ω=⟨ℤ∣⟩. Note that this does not make equality the same as identity: ⟨1∣1⟩=⟨∣⟩, for instance.

It can be shown that N is “sandwiched” between the elements of NL and NR: it is larger than any element of NL and smaller than any element of NR.

In general, ⟨a∣b⟩ is the simplest number between a and b. This can be easily used to define the dyadic fractions: for any integer a, a+12=⟨a∣a+1⟩. Then 12=⟨0∣1⟩, 14=⟨0∣12⟩, and so on. This can then be used to locate non-dyadic fractions by pinning them between a left part which gets infinitely close from below and a right part which gets infinitely close from above.

Ordinal arithmetic can be defined starting with ω as defined above and adding numbers such as ⟨ω∣⟩=ω+1 and so on. Similarly, a starting infinitesimal can be found as ⟨0∣1,12,14⁢…⟩=1ω, and again more can be developed from there.